Question

\[4.0{\text{ }}g\] of gas occupies \[22.4{\text{ }}L{\text{ }}at{\text{ }}NTP\]. The specific heat capacity of the gas at constant volume is \[5.0{\text{ }}J/K\]. If the speed of the sound in this gas at NTP is \[952{\text{ }}m/s\], then the heat capacity at constant pressure is.

\[\left( {Take{\text{ }}gas{\text{ }}constant{\text{ }}R = 8.34{\text{ }}J/K/mol} \right)\]

A. \[8.5J{K^{ - 1}}mo{l^{ - 1}}\]
B. \[8.0J{K^{ - 1}}mo{l^{ - 1}}\]
C. \[87.5J{K^{ - 1}}mo{l^{ - 1}}\]
D. \[7.0J{K^{ - 1}}mo{l^{ - 1}}\]

Answer 1

Hint:

The speed of sound is derived from the kinetic theory of gases in the following way,
\[v = \sqrt {\dfrac{{\gamma RT}}{m}} \]

Where, V is the speed, \[\gamma \] is the adiabatic index, R is the gas constant, T is the temperature and M is the molecular mass.

The adiabatic index is the ratio of specific heat at constant pressure to the specific heat at constant volume.

A gas is said to be containing one mol if it occupies \[22.4{\text{ }}L{\text{ }}at {\text{ }}NTP\].


Complete step by step solution:

Given at Normal Temperature and Pressure, \[a{\text{ }}4.0{\text{ }}g\]gas occupies \[22.4{\text{ }}L\] .

This implies that the moles of gas is 1 mol\[.\left( {Definition{\text{ }}of{\text{ }}NTP} \right)\]

Molar mass of the gas \[m{\text{ }} = {\text{ }}4{\text{ }}gm{\text{ }} = 4 \times {10^{ - 3}}kg\]

Given that, speed of sound is \[952{\text{ }}m/s,{\text{ }}T{\text{ }} = {\text{ }}273{\text{ }}K\]

Using these in the equation of sound,

\[\begin{gathered}
  v = \sqrt {\dfrac{{\gamma RT}}{m}} \\
   = > {v^2} = \dfrac{{\gamma RT}}{m} \\
   = > {(952)^2} = \dfrac{{\gamma \times 8.3 \times 273}}{{4 \times {{10}^{ - 3}}}} \\
   = > \gamma = \dfrac{{{{(952)}^2} \times 4 \times {{10}^{ - 3}}}}{{8.3 \times 273}} \\
   = > \gamma = 8/5 \\
\end{gathered} \]

\[\gamma = \dfrac{{{C_p}}}{{{C_v}}}\]
We know that ,

Given that, the specific heat capacity of the gas at constant volume is \[5.0{\text{ }}J/K/mol\]

I.e., \[{C_v} = 5.0J/K/mol\]

Using this value of and \[\gamma = 8/5\]\[\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\] from above, we get,

\[\dfrac{8}{5} = \dfrac{{{C_p}}}{5}\]

\[{C_p} = 8J{K^{ - 1}}mo{l^{ - 1}}\]

We find that the specific heat at constant pressure is \[8{\text{ }}J/K/mol\].

Hence the correct option is (B).

Note: that specific heat at constant volume\[/\] pressure has same dimensionality of Gas constant
The sound waves are assumed to travel in an adiabatic medium, and not in an isothermal medium as thought by Newton. Which led to the wrong value of Speed of sound. Please keep this in mind.